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ADDITION OF SIGNED NUMBERS
Jump right to the exercises!
See the best ALGEBRA PINBALL time for this exercise
The concepts for this exercise are summarized below. For a complete discussion,
read the text.
Problems like (-2) + (-3) = -5
and (-3) + 5 = 2 are easy for some people and hard for others.
If they're easy for you, then jump right to the exercises.
Otherwise, read on—and keep in mind that explaining something simple, in words, often ends up sounding
very complicated!
The phrase signed numbers refers to numbers that can be
either positive (like 5) or negative (like -5).
That is, signed numbers are allowed
to have a minus sign.
Every real number can be interpreted in two ways:
- as a position on a number line;
- as a movement.
Both of these interpretations—position and movement—are used when learning to add
and subtract signed numbers.
(Once you know how to do it, the process will become automatic and you probably won't think
about the position and movement stuff.)
NUMBERS AS POSITION:
The number 3 can mean: go to position 3 on the number line.
The number -3 can mean: go to position -3 on the number line.
NUMBERS AS MOVEMENT:
Positive numbers can indicate movement to the right. For example, 3 can mean "move 3 units to the right".
Negative numbers can indicate movement to the left. For example, -3 can mean "move 3 units to the left".
When you add a negative number, you should put it in parentheses, unless it comes first.
For example, the sum of -3 and -1 should be written as -3 + (-1) .
(Recall that the word sum refers to an addition problem.)
Every number has a size (its distance from zero).
Every nonzero number has a sign (positive or negative).
For example:
The number 3 : its size is 3, and its sign is positive.
The number -3 : its size is 3, and its sign is negative.
In the movement interpretation of a real number, the size tells us how far to move, and
the sign tells us which direction to move.
Now we're ready to combine the position and
movement ideas in an addition problem.
The process is illustrated
first with an example:
Consider the problem 2 + (-3) + 5 .
The first number, 2 , indicates a position. Go to 2 on the number line.
Adding a negative number indicates movement to the left. Thus, adding -3 says to move 3 units to the left.
Adding a positive number indicates movement to the right. Thus, adding 5 says to move 5 units to the right.
You end up at position 4 . Thus, 2 + (-3) + 5 = 4 .
Or, you can always start at zero!
That is, write 2 + (-3) + 5 as 0 + 2 + (-3) + 5 .
Start at 0 , move 2 to the right, 3 to the left, and 5 to the right, ending up at 4 .
You should understand both interpretations, but in practice you can use whichever is more natural to you.
The start at zero interpretation is used in the following discussion.
You probably don't want to be drawing number lines every time you need to do an
addition of signed numbers problem. The good news is that every problem boils down to either
an addition problem or a subtraction problem, which can be done efficiently in your head.
Keep reading!
When you add numbers with the
same signs (both positive or both negative), then in your
head you do an addition problem.
Here are two examples:
2 + 3 : Start at zero. Move to the right 2, then to the right 3. End up at 5.
Thus 2 + 3 = 5 .
-2 + (-3) : Start at zero. Move to the left 2, then to the left 3.
The total
distance moved is 2 + 3 = 5 . You moved to the left, so you end up at -5. Thus,
-2 + (-3) = -5 .
Notice that in both of these problems, you do an addition problem in your head, which gives the total distance moved.
If you always move to the right, the final answer is positive.
If you always move to the left, the final answer is negative.
When you add numbers with different signs (one positive, one negative), then in your head you do a subtraction problem.
Here are two examples:
2 + (-3) : Start at zero. Move 2 to the right and 3 to the left.
You moved more to
the left—how much more? 3 - 2 = 1 . So you end up at -1.
Thus, 2 + (-3) = -1 .
3 + (-2) : Start at zero. Move 3 to the right and 2 to the left.
You moved
more to the right—how much more? 3 - 2 = 1 . So you end up at 1.
Thus, 3 + (-2) = 1 .
The mental process is this:
Once you recognize that you're adding numbers with different signs,
throw away (for the moment) all the signs,
take the bigger number, and subtract the smaller number.
This gives you the
net distance traveled.
If you move farther to the right, your answer is positive.
If you move farther to the left, your answer is negative.
Notice that when you add numbers with different signs, in your head you do a subtraction problem.
FIVE-STEP PROCESS FOR ADDING TWO SIGNED NUMBERS:
When you add two signed numbers, you can follow this five step process.
As you read through these steps, think of applying these questions to the problem 2 + (-3) :
- Step 1: What numbers are being added? (Answer: 2 and -3)
- Step 2: Do these numbers have the same sign or different signs? (Answer: different signs)
- Step 3: In your head, will you be doing an addition or subtraction problem? (Answer: subtraction problem)
- Step 4: Do the appropriate addition or subtraction problem.
(Answer: Throw away the signs, leaving you with 2 and 3. Subtract the smaller from the larger: 3 - 2 = 1 )
- Step 5: Is your answer positive or negative?
(Answer: The bigger number is negative. So 2 + (-3) = -1 )
If there are more than two numbers being added,
just turn it into a two-number problem in the first step,
by combining the positive and negative numbers separately, like this:
-3 + 5 + (-2) + 1 + 4 + (-6)
= ( -3 + (-2) + (-6) ) + ( 5 + 1 + 4 ) (re-group, re-order, to combine negative and positive separately)
= -11 + 10
= -1
Here, you will practice addition problems of the form "x + y"
where x and y can be any of these numbers: -10, -9, -8, ..., -1, 0, 1, ..., 8, 9, 10 .
About half of the problems will involve variables!
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.